Student Solutions Manual for Differential Equations and Linear Algebra

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Student Solutions Manual for Differential Equations and Linear Algebra

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Table of Contents

  1. First-Order Differential Equations
    • 1.1 Differential Equations and Mathematical Models
    • 1.2 Integrals as General and Particular Solutions
    • 1.3 Slope Fields and Solution Curves
    • 1.4 Separable Equations and Applications
    • 1.5 Linear First-Order Equations
    • 1.6 Substitution Methods and Exact Equations
  2. Mathematical Models and Numerical Methods
    • 2.1 Population Models
    • 2.2 Equilibrium Solutions and Stability
    • 2.3 Acceleration–Velocity Models
    • 2.4 Numerical Approximation: Euler’s Method
    • 2.5 A Closer Look at the Euler Method
    • 2.6 The Runge–Kutta Method
  3. Linear Systems and Matrices
    • 3.1 Introduction to Linear Systems
    • 3.2 Matrices and Gaussian Elimination
    • 3.3 Reduced Row-Echelon Matrices
    • 3.4 Matrix Operations
    • 3.5 Inverses of Matrices
    • 3.6 Determinants
    • 3.7 Linear Equations and Curve Fitting
  4. Vector Spaces
    • 4.1 The Vector Space R3
    • 4.2 The Vector Space Rn and Subspaces
    • 4.3 Linear Combinations and Independence of Vectors
    • 4.4 Bases and Dimension for Vector Spaces
    • 4.5 Row and Column Spaces
    • 4.6 Orthogonal Vectors in Rn
    • 4.7 General Vector Spaces
  5. Higher-Order Linear Differential Equations
    • 5.1 Introduction: Second-Order Linear Equations
    • 5.2 General Solutions of Linear Equations
    • 5.3 Homogeneous Equations with Constant Coefficients
    • 5.4 Mechanical Vibrations
    • 5.5 Nonhomogeneous Equations and Undetermined Coefficients
    • 5.6 Forced Oscillations and Resonance
  6. Eigenvalues and Eigenvectors
    • 6.1 Introduction to Eigenvalues
    • 6.2 Diagonalization of Matrices
    • 6.3 Applications Involving Powers of Matrices
  7. Linear Systems of Differential Equations
    • 7.1 First-Order Systems and Applications
    • 7.2 Matrices and Linear Systems
    • 7.3 The Eigenvalue Method for Linear Systems
    • 7.4 A Gallery of Solution Curves of Linear Systems
    • 7.5 Second-Order Systems and Mechanical Applications
    • 7.6 Multiple Eigenvalue Solutions
    • 7.7 Numerical Methods for Systems
  8. Matrix Exponential Methods
    • 8.1 Matrix Exponentials and Linear Systems
    • 8.2 Nonhomogeneous Linear Systems
    • 8.3 Spectral Decomposition Methods
  9. Nonlinear Systems and Phenomena
    • 9.1 Stability and the Phase Plane
    • 9.2 Linear and Almost Linear Systems
    • 9.3 Ecological Models: Predators and Competitors
    • 9.4 Nonlinear Mechanical Systems
  10. Laplace Transform Methods
    • 10.1 Laplace Transforms and Inverse Transforms
    • 10.2 Transformation of Initial Value Problems
    • 10.3 Translation and Partial Fractions
    • 10.4 Derivatives, Integrals, and Products of Transforms
    • 10.5 Periodic and Piecewise Continuous Input Functions
  11. Power Series Methods
    • 11.1 Introduction and Review of Power Series
    • 11.2 Power Series Solutions
    • 11.3 Frobenius Series Solutions
    • 11.4 Bessel Functions

Appendix A: Existence and Uniqueness of Solutions

Appendix B: Theory of Determinants

APPLICATION MODULES

The modules listed below follow the indicated sections in the text. Most provide computing projects that illustrate the corresponding text sections. Many of these modules are enhanced by the supplementary material found at the new Expanded Applications website.

  • 1.3 Computer-Generated Slope Fields and Solution Curves
  • 1.4 The Logistic Equation
  • 1.5 Indoor Temperature Oscillations
  • 1.6 Computer Algebra Solutions
  • 2.1 Logistic Modeling of Population Data
  • 2.3 Rocket Propulsion
  • 2.4 Implementing Euler’s Method
  • 2.5 Improved Euler Implementation
  • 2.6 Runge-Kutta Implementation
  • 3.2 Automated Row Operations
  • 3.3 Automated Row Reduction
  • 3.5 Automated Solution of Linear Systems
  • 5.1 Plotting Second-Order Solution Families
  • 5.2 Plotting Third-Order Solution Families
  • 5.3 Approximate Solutions of Linear Equations
  • 5.5 Automated Variation of Parameters
  • 5.6 Forced Vibrations and Resonance
  • 7.1 Gravitation and Kepler’s Laws of Planetary Motion
  • 7.3 Automatic Calculation of Eigenvalues and Eigenvectors
  • 7.4 Dynamic Phase Plane Graphics
  • 7.5 Earthquake-Induced Vibrations of Multistory Buildings
  • 7.6 Defective Eigenvalues and Generalized Eigenvectors
  • 7.7 Comets and Spacecraft
  • 8.1 Automated Matrix Exponential Solutions
  • 8.2 Automated Variation of Parameters
  • 9.1 Phase Portraits and First-Order Equations
  • 9.2 Phase Portraits of Almost Linear Systems
  • 9.3 Your Own Wildlife Conservation Preserve
  • 9.4 The Rayleigh and van der Pol Equations

About the Book

 

Build on conceptual ideas and the use of applications and projects to involve students in active problem solving experiences

  • 600 computer-generated figures show students vivid pictures of direction fields, solution curves, and phase plane portraits that bring symbolic solutions of differential equations to life.
    • NEW! Nearly 80 new figures illustrating interactive computer applications
  • Application modules follow key sections throughout. Most outline “technology neutral” investigations illustrating the use of technical computing systems and seek to actively engage students in the application of new technology.
  • A fresh numerical emphasis is afforded by the early introduction of numerical solution techniques in Chapter 2 (on mathematical models and numerical methods). Includes numerical algorithms presented in parallel fashion for systems ranging from graphing calculators to MATLAB.
  • A conceptual perspective shaped by the availability of computational aids permits a leaner and more streamlined coverage of certain traditional symbolic methods (like exact equations and variation of parameters).

Widely used technical computing environments used by practicing engineers and scientists are heavily integrated with content

  • NEW! Now includes Python, an increasingly popular computer platform that is freely available on the internet and an all-purpose scientific computing environment.
  • NEW! The online Expanded Applications companion website enhances the text’s applications material through technology, and provides detailed coverage of Maple, Mathematica, and MATLAB techniques.

For courses in Differential Equations and Linear Algebra.

 

Now available with MyLab Math 

The right balance between concepts, visualization, applications, and skills – now available with MyLab Math

Differential Equations and Linear Algebra provides the conceptual development and geometric visualization of a modern differential equations and linear algebra course that is essential to science and engineering students. It balances traditional manual methods with the new, computer-based methods that illuminate qualitative phenomena – a comprehensive approach that makes accessible a wider range of more realistic applications.

The book combines core topics in elementary differential equations with concepts and methods of elementary linear algebra. It starts and ends with discussions of mathematical modeling of real-world phenomena, evident in figures, examples, problems, and applications throughout. 

For the first time, MyLab Math is available for this text, providing online homework with immediate feedback, the complete eText, and more. Additionally, new presentation slides created by author David Calvis are available in Beamer (LaTeX) and PDF formats. The slides are ideal for classroom lectures and student review, and combined with Calvis’ superlative instructional videos offer a level of support not found in any other Differential Equations course.

Personalize learning with MyLab Math

MyLab Math is the teaching and learning platform that empowers you to reach every student. By combining trusted author content with digital tools and a flexible platform, MyLab Math personalizes the learning experience and improves results for each student. 

0136743641 / 9780136743644  MYLAB MATH DIGITAL UPDATE WITH PEARSON ETEXT — ACCESS CARD — FOR DIFFERENTIAL EQUATIONS AND LINEAR ALGEBRA, 4/e

C. Henry Edwards is emeritus professor of mathematics at the University of Georgia. He earned his Ph.D. at the University of Tennessee in 1960, and recently retired after 40 years of classroom teaching (including calculus or differential equations almost every term) at the universities of Tennessee, Wisconsin, and Georgia, with a brief interlude at the Institute for Advanced Study (Princeton) as an Alfred P. Sloan Research Fellow. He has received numerous teaching awards, including the University of Georgia’s honoratus medal in 1983 (for sustained excellence in honors teaching), its Josiah Meigs award in 1991 (the institution’s highest award for teaching), and the 1997 statewide Georgia Regents award for research university faculty teaching excellence. His scholarly career has ranged from research and dissertation direction in topology to the history of mathematics to computing and technology in the teaching and applications of mathematics. In addition to being author or co-author of calculus, advanced calculus, linear algebra, and differential equations textbooks, he is well-known to calculus instructors as author of The Historical Development of the Calculus (Springer-Verlag, 1979). During the 1990s he served as a principal investigator on three NSF-supported projects: (1) A school mathematics project including Maple for beginning algebra students, (2) A Calculus-with-Mathematica program, and (3) A MATLAB-based computer lab project for numerical analysis and differential equations students.

 

David E. Penney, University of Georgia, completed his Ph.D. at Tulane University in 1965 (under the direction of Prof. L. Bruce Treybig) while teaching at the University of New Orleans. Earlier he had worked in experimental biophysics at Tulane University and the Veteran’s Administration Hospital in New Orleans under the direction of Robert Dixon McAfee, where Dr. McAfee’s research team’s primary focus was on the active transport of sodium ions by biological membranes. Penney’s primary contribution here was the development of a mathematical model (using simultaneous ordinary differential equations) for the metabolic phenomena regulating such transport, with potential future applications in kidney physiology, management of hypertension, and treatment of congestive heart failure. He also designed and constructed servomechanisms for the accurate monitoring of ion transport, a phenomenon involving the measurement of potentials in microvolts at impedances of millions of megohms. Penney began teaching calculus at Tulane in 1957 and taught that course almost every term with enthusiasm and distinction until his retirement at the end of the last millennium. During his tenure at the University of Georgia he received numerous University-wide teaching awards as well as directing several doctoral dissertations and seven undergraduate research projects. He is the author of research papers in number theory and topology and is the author or co-author of textbooks on calculus, computer programming, differential equations, linear algebra, and liberal arts mathematics.

About the Book

 

  • New text and graphics have been inserted in a number of sections, to enhance student understanding of the subject matter.
    • Nearly 80 new figures illustrating interactive computer applications
  • Now includes Python, an increasingly popular computer platform that is freely available on the internet and an all-purpose scientific computing environment.
  • The online Expanded Applications companion website enhances the text’s applications material through technology, and provides detailed coverage of Maple, Mathematica, and MATLAB techniques.

 

Content Updates

 

  • Chapter 7 has a new section devoted to the construction of a “gallery” of phase plane portraits illustrating all the possible geometric behaviors of solutions of the 2-dimensional linear system x’ = Ax.
  • Chapter 9 now contains a new biological application whichincludes a substantial investigation of the nonlinear FitzHugh-Nagumo equations of neuroscience.

For courses in Differential Equations and Linear Algebra.

The right balance between concepts, visualization, applications, and skills 
Differential Equations and Linear Algebra provides the conceptual development and geometric visualization of a modern differential equations and linear algebra course that is essential to science and engineering students. It balances traditional manual methods with the new, computer-based methods that illuminate qualitative phenomena — a comprehensive approach that makes accessible a wider range of more realistic applications.

The book combines core topics in elementary differential equations with concepts and methods of elementary linear algebra. It starts and ends with discussions of mathematical modeling of real-world phenomena, evident in figures, examples, problems, and applications throughout. 

Extend learning beyond the classroom
Pearson eText is an easy-to-use digital textbook that students can purchase on their own or you can assign for your course. It lets students read, highlight, and take notes all in one place. The mobile app lets students learn on the go, offline or online. Creating a course allows you to schedule readings, view reading analytics, and share your own notes with students, motivating them to keep reading, and keep learning. Learn more about Pearson eText.

Additional information

Dimensions 1.40 × 8.55 × 10.80 in
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Subjects

mathematics, higher education, Calculus, Applied & Advanced Math, Advanced Math, Differential Equations and Linear Alg